CN109946029B - Multi-frequency fatigue test method for turbocharger - Google Patents

Multi-frequency fatigue test method for turbocharger Download PDF

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CN109946029B
CN109946029B CN201910133457.2A CN201910133457A CN109946029B CN 109946029 B CN109946029 B CN 109946029B CN 201910133457 A CN201910133457 A CN 201910133457A CN 109946029 B CN109946029 B CN 109946029B
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blade
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CN109946029A (en
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李伟
邢鑫鑫
杲宁
孙锐
周思奇
李萌
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Beijing Institute of Technology BIT
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Abstract

The invention discloses a multi-frequency fatigue test method of a turbocharger, which is different from a common multi-frequency test method of the turbocharger by using a tuning mass under the condition of not changing the requirement of a fatigue test, and adopts two or more eigenfrequencies to excite a turbine rotor blade of the turbocharger so as to reduce the fatigue test time, weaken the complexity of problems and ensure that the test result has higher fidelity. In addition, from the view point of not using the tuning mass, the blades are simultaneously excited by using two or more eigenfrequency combinations, and the actual space damage distribution of the turbine rotor blades of the turbocharger is simulated more realistically by tuning the amplitude of the exciting force.

Description

Multi-frequency fatigue test method for turbocharger
Technical Field
The invention relates to a blade fatigue test method, in particular to a multi-frequency fatigue test method for a turbocharger turbine rotor blade, and belongs to the field of test and test.
Background
Rotor blades are one of the critical components of a turbine, which are required to have high reliability, and the service life of the blades depends to a large extent on their fatigue life. Therefore, fatigue testing of turbine rotor blades is essential.
The existing turbine blade fatigue test method is mainly used for carrying out full-size blade tests, and exciting the blades based on a typical rotating mass to achieve the aim of resonance excitation, so that the fatigue test of the blades is realized. The current test method only uses the lowest eigenfrequency to cause virtual damage, and the additional tuning mass is used to restore the real damage distribution, even if the amplitude amplification technology is adopted to carry out fatigue test, much time is still spent. In prior fatigue testing for turbine rotor blades, the use of tuned masses increased the complexity of the fatigue test, which decreased the natural frequency of the blade, thereby increasing the test time.
Therefore, a new fatigue test method for simulating the actual spatial damage distribution more realistically by tuning the amplitude of the excitation force by simultaneously exciting the blades by using two or more eigenfrequencies in combination from the viewpoint of reducing the fatigue test time and ensuring higher fidelity without using tuning mass is urgently needed to be established.
Disclosure of Invention
In view of the above, the invention provides a multi-frequency fatigue test method for a turbocharger, which adopts a blade fixing structure, a blade to be tested and an eigenfrequency excitation structure for carrying out a test, wherein one end of the blade to be tested is fixed on the fixing structure, and the eigenfrequency excitation structure is an actuator arranged below the blade to be tested or at least two vibration exciters arranged above the blade to be tested;
the method comprises the following specific steps:
step 1: discretizing the blade to simplify the problem of the whole blade. Discretizing the blade by adopting a two-dimensional cluster mass model to obtain a discretized mass model of the blade, wherein
Figure BDA0001976182320000021
For the elements of the basic quality matrix,
Figure BDA0001976182320000022
for the k-th additional tuning quality matrix,
Figure BDA0001976182320000023
for the kth tuning element or excitation mass matrix, ei (z) is the bending stiffness in sway, c (z) is the chord length, m (z) is the blade mass per unit length, the model is generated in a multi-norm numerical computing environment matlabv.15a, and consists of 100 nodes n, L is the total length of the blade, equidistant at △ z L/n, and △ z is the equidistant distance between adjacent nodes, each node has one degree of translational freedom in the x-direction, the dynamic response of the lumped mass system under dual-frequency excitation shown by the model is described as follows, and a nonlinear second order differential equation expressed by a dot symbol:
Figure BDA0001976182320000024
in the formula: m is a total mass matrix, D is a Rayleigh damping coefficient, A is a pneumatic damping matrix, K is a stiffness matrix, f1 and 2 are node exciting force amplitude vectors, omega1And omega2Is firstAnd the angular natural characteristic frequency of the second mode, t being time,
Figure BDA0001976182320000025
is the absolute value of the component velocity vector,
Figure BDA0001976182320000026
in the form of a velocity vector, the velocity vector,
Figure BDA0001976182320000027
is an acceleration vector;
step 2: and solving the rigidity matrix by using the Euler-Bernoulli theorem. Using first order Euler-Bernoulli Beam theory according to equation (2), for a symmetric compliance matrix (z)i<zj) Each element of the upper triangle of (1) is processed by semi-resolving, where z is a spanwise coordinate from the base, z is a distance from the basei，jFor the spanwise coordinates of the ith and j nodes, using the principle of imaginary work, equation (2) represents the deformation of node i caused by the unit load applied by node j, equation (2) is obtained by numerical integration using the Simpson's rule, and the inverse of the compliance matrix yields the stiffness matrix shown in equation (3),
Figure BDA0001976182320000028
K＝S-1(3)
in the formula, SijIs a compliance matrix element, S is a compliance matrix,
and step 3:
the total mass matrix is solved using the correlation function. Elements of the basic mass matrix of the blade
Figure BDA00019761823200000313
Can be obtained by the following formula:
Figure BDA0001976182320000031
in the formula (I), the compound is shown in the specification,ijfor the Kronecker function, the lumped mass model formula requires that extra (tuning or excitation) be addedThe quality is distributed discretely at its nodes by distributing the kth additional quality between its two adjacent nodes i and i +1
Figure BDA0001976182320000032
Equation (5) is used to calculate the corresponding distribution quality matrix
Figure BDA0001976182320000033
Figure BDA0001976182320000034
In the formula, zkFor the coordinates of the kth additional mass,
Figure BDA0001976182320000035
and
Figure BDA0001976182320000036
is an n-dimensional orthogonal basis vector,
Figure BDA0001976182320000037
and
Figure BDA0001976182320000038
is a transposed matrix of i-dimensional orthogonal basis vectors, zkFor the kth spanwise coordinate, the total mass matrix M can thus be obtained:
Figure BDA0001976182320000039
in the formula (I), the compound is shown in the specification,
Figure BDA00019761823200000310
in order to be a basic quality matrix,
Figure BDA00019761823200000311
for the kth additional tuning or excitation quality matrix, ntIn order to tune the mass number,
and 4, step 4:
the influence of air damping is taken into account, and the influence of damping is taken into account in the calculation process. Rayleigh damping is expressed in its most common form, as shown in detail below,
D＝aM+bK (7)
in the above formula, a is 0 and b is 0.0156, a and b are rayleigh damping coefficients,
element A of the aerodynamic damping matrixijCan be obtained by the formula (8) wherein C is perpendicular to the flow of the flat platedAs a conservative approximation representing a complex aerodynamic situation:
Figure BDA00019761823200000312
where ρ is the air density at ambient room temperature, C (z) is the chord length, CdIs an aerodynamic drag coefficient of an angle of attack of 90 degrees,
Figure BDA0001976182320000041
Figure BDA0001976182320000042
the corresponding eigenvectors of the defined model shape are derived from equation (10) by solving the well-known generalized eigenvalue problem set forth in equation (9), where φiN-dimensional eigenmode vectors for the ith mode: and 5:
the calculation of the additional parameter, the static bending moment distribution caused by the blade self-weight is calculated by a two-step procedure, first calculating the shear force Q (z) using equation (11) when the boundary condition shear force Q (L) of the tip is 0:
Figure BDA0001976182320000043
where g is the gravitational acceleration and M(s) is the blade mass per unit length, and then the moment distribution M is obtained using equation (12)S(z) and a boundary condition Ms(L)＝0：
Figure BDA0001976182320000044
In the formula, MS(z) a bending moment of oscillation caused by its own weight, MS(L) is the bending moment of oscillation at L length due to its own weight, Q(s) is the shear force at length s,
Figure BDA0001976182320000045
for the k-th tuning mass,
Figure BDA0001976182320000046
the spanwise coordinate of the kth additional mass, the static bending moment distribution caused by the dead weight of the external mass, can be obtained by equation (13), and the generated static bending moment can be obtained by equation (14) using the superposition principle:
Figure BDA0001976182320000047
Figure BDA0001976182320000049
in the formula, Mt(z) is the bending moment created by the tuned mass,
Figure BDA00019761823200000410
for the average (static) bending moment, the node bending moment history M (z, t) used for damage calculation is represented by equation (15), where the node curvature is approximated by the central difference of the second derivative of the node displacement history x (z, t):
Figure BDA0001976182320000051
step 6:
and solving the degree of mixing of the double excitation modes. Limiting the peak strain amplitude to an empirical sensor type specific threshold, the root mean square strain threshold given by equation (16)
Figure BDA0001976182320000052
Used as constraints in the optimization process discussed later:
Figure BDA0001976182320000053
wherein the time span T2-T1The strain history (z, t) of the end fiber corresponding to the steady-state portion of the time history and the flange, which yields bernoulli beams through the strain-moment relationship of first order Euler, as shown in equation (17):
Figure BDA0001976182320000054
in the formula, xc(z) is the distance between the terminal fiber and the bending axis.
In the case of dual-frequency excitation, the ratio between the two force amplitudes is measured by the modal mixing degree, defined here as
Figure BDA0001976182320000055
Where ψ is a degree of mixing of the dual excitation modes, where ψ ═ 1 means pure mode-1 excitation, ψ ═ 0 means pure mode-2 excitation, and F is1And F2In order to obtain the amplitude of the exciting force,
and 7:
in the multi-frequency fatigue experiment method, a relational expression of design vector variables and related parameters is established, and the nonlinear constraint of final fatigue failure is established from the angle of energy and actual damage distribution.
In the dual-frequency method, the design variable vector q consists of two excitation forces: excitation position and scale factor, given by equation (19):
Figure BDA0001976182320000061
wherein w is a scaling factor, the vector of the total node cycle number and the total test time and total dissipated energy are expressed by the formulas (20) to (24)Given that the objective function of the multi-frequency method is in accordance with equation (25), the node cycle number in the case of multi-frequency varies with the blade variation, and therefore, the non-linear constraint in equation (28) requires that the minimum node cycle number is equal to or greater than 3 × 106And (3) cyclic reference:
Figure BDA0001976182320000062
Ttot＝w(T2-T1) (21)
Figure BDA0001976182320000063
Figure BDA0001976182320000064
Figure BDA0001976182320000065
Figure BDA0001976182320000066
in the formula, NrsIs the number of cycles in unit rs, ntotAs vector of total node cycle number, TtotFor total test time, EtotFor total energy consumption, D is the actual damage index (fatigue failure is reached when D ═ 1), EaerFor aerodynamic energy dissipation, EstrFor structural energy dissipation, DtIs a target damage index
Minimum f (q) obedience
3×106-ntot(q)＝0 (26)
Figure BDA0001976182320000067
Figure BDA0001976182320000068
In the formula (I), the compound is shown in the specification,
Figure BDA0001976182320000069
root mean square strain value, x, of time history100The distance between the end fiber and the bending axis was 100 mm.
Drawings
In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the provided drawings without creative efforts,
FIG. 1a is a schematic diagram of a conventional single frequency fatigue testing apparatus;
FIG. 1b is a drawing of a multi-frequency fatigue testing apparatus using rotating mass excitation;
FIG. 1c is a schematic representation of a surface drive that may be used for both single frequency and multi-frequency excitation;
FIG. 2a is a schematic view of the associated nodal plan area for aerodynamic damping;
FIG. 2b is a schematic illustration of the associated segment mass;
FIG. 2c is a discretized mass model of the blade.
Wherein, 1-fixed structure, 2-blade to be measured, 3-vibration exciter, 4-actuator, 5 relevant node plane area.
Detailed Description
In the following, the technical solutions in the embodiments of the present invention will be clearly and completely described, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, rather than all embodiments, and all other embodiments obtained by those skilled in the art without any inventive work based on the embodiments of the present invention belong to the protection scope of the present invention,
FIGS. 1 a-1 b illustrate different fatigue testing apparatus and methods for turbine blades, in which
Figure BDA0001976182320000071
For the kth tuning mass (k ═ 1,2 … …),
Figure BDA0001976182320000072
spanwise coordinate, ω, for the kth additional massiIs the angular natural characteristic frequency (i ═ 1,2 … …), F, of the i-th mode1,2Is the amplitude of the exciting force. Fig. 1(a) shows a single-frequency fatigue testing apparatus and method which are most commonly used in industry, and a multi-frequency fatigue testing apparatus for a new turbine device blade is constructed based on an existing single-frequency fatigue testing apparatus. Fig. 1(b) and (c) show two applications of the multi-frequency test method.
The specific test steps are as follows:
step 1:
discretizing the blade to simplify the problem of the whole blade. Based on the existing multi-frequency test device, a two-dimensional cluster mass model is adopted to carry out discretization processing on the blade, and a discretization mass model of the blade is obtained. As shown in FIG. 2, in which
Figure BDA0001976182320000081
For the elements of the basic quality matrix,
Figure BDA0001976182320000082
for the k-th additional tuning quality matrix,
Figure BDA0001976182320000083
the model is generated in a multi-norm numerical computing environment Matlab v.15a, and consists of n-100 nodes, L is the total length of the blade, equidistant at △ z-L/n, and △ z is the equidistant distance between adjacent nodes, each node has one translational degree of freedom in the x direction.
Figure BDA0001976182320000084
In the formula: m is a total mass matrix, D is a Rayleigh damping coefficient, A is a pneumatic damping matrix, K is a stiffness matrix, f1,2Is a node excitation force amplitude vector, omega1And omega2The angular eigenfrequencies of the first and second modes, t is time,
Figure BDA0001976182320000085
is the absolute value of the component velocity vector,
Figure BDA0001976182320000086
in the form of a velocity vector, the velocity vector,
Figure BDA0001976182320000087
is an acceleration vector;
step 2:
and solving the rigidity matrix by using the Euler-Bernoulli theorem. Using first order Euler-Bernoulli Beam theory according to equation (2), for a symmetric compliance matrix (z)i<zj) Each element of the upper triangle of (1) is processed by semi-resolving, where z is a spanwise coordinate from the base, z is a distance from the basei，jThe spanwise coordinates of the ith and j nodes. Using the principle of imaginary work, equation (2) represents the deformation of the node i caused by the unit load applied by the node j. Equation (2) is obtained by numerical integration using the Simpson's rule, and the inverse of the compliance matrix may result in the stiffness matrix shown in equation (3).
Figure BDA0001976182320000088
K＝S-1(3)
In the formula, SijIs a compliance matrix element, S is a compliance matrix,
and step 3:
the total mass matrix is solved using the correlation function. Elements of the basic mass matrix of the blade
Figure BDA0001976182320000091
Can be obtained by the following formula:
Figure BDA0001976182320000092
in the formula (I), the compound is shown in the specification,ijthe lumped mass model formula requires that the extra (tuning or excitation) mass be distributed discretely across its nodes for the Kronecker function. By distributing the kth additional quality between its two neighboring nodes i and i +1
Figure BDA0001976182320000093
Equation (5) is used to calculate the corresponding distribution quality matrix,
Figure BDA0001976182320000094
in the formula, zkFor the coordinates of the kth additional mass,
Figure BDA0001976182320000095
and
Figure BDA0001976182320000096
is an n-dimensional orthogonal basis vector,
Figure BDA0001976182320000097
and
Figure BDA0001976182320000098
is a transposed matrix of i-dimensional orthogonal basis vectors, zkFor the kth spanwise coordinate, the total mass matrix M can thus be obtained:
Figure BDA0001976182320000099
in the formula (I), the compound is shown in the specification,
Figure BDA00019761823200000910
in order to be a basic quality matrix,
Figure BDA00019761823200000911
for the kth additional tuning or excitation quality matrix, ntTo tune the mass number.
And 4, step 4:
the influence of air damping is taken into account, and the influence of damping is taken into account in the calculation process. Rayleigh damping is expressed in its most common form, as shown in detail below,
D＝aM+bK (7)
in the above formula, a is 0 and b is 0.0156, and a and b are rayleigh damping coefficients.
Element A of the aerodynamic damping matrixijCan be obtained by the formula (8) wherein C is perpendicular to the flow of the flat platedAs a conservative approximation representing a complex aerodynamic situation:
Figure BDA0001976182320000101
where ρ is the air density at ambient room temperature, C (z) is the chord length, CdIs the aerodynamic drag coefficient at an angle of attack of 90 deg..
Figure BDA0001976182320000102
Figure BDA0001976182320000103
The corresponding eigenvectors of the defined model shape are derived from equation (10) by solving the well-known generalized eigenvalue problem set forth in equation (9), where φiN-dimensional eigenmode vectors for the ith mode: and 5:
and the influence of static bending moment is considered, so that the problem solving process is ensured to have higher reliability. Additional parameter calculations were performed and the static bending moment distribution caused by the blade self-weight was calculated by a two-step procedure.
First, the shear force Q (z) is calculated using equation (11) when the boundary condition shear force Q (L) of the tip is 0:
Figure BDA0001976182320000104
where g is the gravitational acceleration and M(s) is the blade mass per unit length, and then the moment distribution M is obtained using equation (12)S(z) and a boundary condition Ms(L)＝0：
Figure BDA0001976182320000105
In the formula, MS(z) a bending moment of oscillation caused by its own weight, MS(L) is the sway bending moment caused by the self weight at length L, q(s) is the shear force at length s, the static bending moment distribution caused by the self weight of the external mass can be obtained by equation (13), and the resulting static bending moment can be obtained by equation (14) using the superposition principle:
Figure BDA0001976182320000106
Figure BDA0001976182320000111
in the formula (I), the compound is shown in the specification,
Figure BDA0001976182320000112
for the k-th tuning mass,
Figure BDA0001976182320000113
spanwise coordinate of the kth additional mass, Mt(z) is the bending moment created by the tuned mass,
Figure BDA0001976182320000114
is the average (static) bending moment. The node bending moment history M (z, t) used for damage calculation is represented by equation (15), where the node curvature is approximated by the central difference of the second derivative of the node displacement history x (z, t):
Figure BDA0001976182320000115
step 6:
and solving the degree of mixing of the double excitation modes. The life of the strain sensor can affect test time due to test interruptions and equipment replacement. Therefore, the peak strain amplitude is limited to an empirical sensor type specific threshold, the root mean square strain threshold given by equation (16)
Figure BDA0001976182320000116
Used as constraints in the optimization process discussed later:
Figure BDA0001976182320000117
wherein the time span T2-T1The strain history (z, t) of the end fiber corresponding to the steady-state portion of the time history and the flange, which yields bernoulli beams through the strain-moment relationship of first order Euler, as shown in equation (17):
Figure BDA0001976182320000118
in the formula, xc(z) is the distance between the terminal fiber and the bending axis.
In the case of dual-frequency excitation, the ratio between the two force amplitudes is measured by the modal mixing degree, defined here as
Figure BDA0001976182320000119
Where ψ is a degree of mixing of the dual excitation modes, where ψ ═ 1 means pure mode-1 excitation, ψ ═ 0 means pure mode-2 excitation, and F is1And F2In order to obtain the amplitude of the exciting force,
and 7:
in the multi-frequency fatigue experiment method, a relational expression of design vector variables and related parameters is established, and the nonlinear constraint of final fatigue failure is established from the angle of energy and actual damage distribution.
The fatigue test must satisfy four general constraints which can be roughly summarized as a minimum number of cycles, a maximum strain threshold, a spatial damage distribution and a limit of adiabatic heating. In order to find a solution that satisfies the above constraints, a numerical optimization algorithm is required to achieve sufficient accuracy and efficiency. For the current problem, a Matlab optimizer fmincon is used for nonlinear constraint optimization. Fmincon calculates the Hessian matrix by finite difference by default, which is reasonable assuming a small finite difference error due to the semi-analytical formula of the model. In the case of a double frequency, the blade is excited by two excitation forces simultaneously. In the dual-frequency method, the design variable vector q consists of two excitation forces: excitation position and scale factor, given by equation (19):
Figure BDA0001976182320000121
the vector of total node cycle numbers and the total test time and total dissipated energy are given by equations (20) to (24), and the objective function of the multi-frequency method is in accordance with equation (25). the node cycle numbers in the multi-frequency case vary with the blade variation, therefore, the non-linear constraint in equation (28) requires that the minimum node cycle number be equal to or greater than 3 × 106And (3) cyclic reference:
Figure BDA0001976182320000122
Ttot＝w(T2-T1) (21)
Figure BDA0001976182320000123
Figure BDA0001976182320000124
Figure BDA0001976182320000125
Figure BDA0001976182320000131
in the formula, NrsIs the number of cycles in unit rs, ntotAs vector of total node cycle number, TtotFor total test time, EtotFor total energy consumption, D is the actual damage index (fatigue failure is reached when D ═ 1), EaerFor aerodynamic energy dissipation, EstrFor structural energy dissipation, DtIs a target damage index
Minimum f (q) obedience
3×106-ntot(q)＝0 (26)
Figure BDA0001976182320000132
Figure BDA0001976182320000133
In the formula (I), the compound is shown in the specification,
Figure BDA0001976182320000134
root mean square strain value, x, of time history100The distance between the end fiber and the bending axis was 100 mm.
The invention discloses a multi-frequency fatigue test method of a turbocharger, which reduces fatigue test time, ensures higher fidelity, does not use the angle of tuning quality, simultaneously excites blades by utilizing two or more eigenfrequency combinations, and simulates actual spatial damage distribution more vividly by tuning the amplitude of exciting force
In the present specification, the embodiments are described in a progressive manner, each embodiment focuses on the difference from the other embodiments, and the same and similar parts among the embodiments are referred to each other.
The foregoing description of the disclosed embodiments will enable those skilled in the art to make or use the invention, and it will be apparent to those skilled in the art that various modifications to these embodiments may be made, and the general principles defined herein may be applied to other embodiments without departing from the spirit or scope of the invention, and the invention is therefore not to be limited to the embodiments illustrated herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims (1)

1. A multi-frequency fatigue test method of a turbocharger is characterized in that a blade fixing structure, a blade to be tested and an eigenfrequency excitation structure are adopted for testing, one end of the blade to be tested is fixed on the fixing structure, and the eigenfrequency excitation structure is an actuator arranged below the blade to be tested or at least two vibration exciters arranged above the blade to be tested;
the method comprises the following specific steps:
step 1, discretizing the blade, and discretizing the blade by adopting a two-dimensional cluster mass model to obtain a discretized mass model of the blade, wherein
Figure FDA0002424596660000011
For the elements of the basic quality matrix,
Figure FDA0002424596660000012
for the k-th additional tuning quality matrix,
Figure FDA0002424596660000013
for the k-th tuning element or excited mass matrix, where k is the number of times, i and j are the number of rows and columns, i, j, k is 1,2 … …, ei (z) is the bending stiffness of oscillation, c (z) is the chord length, m (z) is the mass of the blade per unit length, the model is generated in a multi-norm numerical calculation environment Matlab v.15a, and consists of n-100 nodes, L is the total length of the blade, and is equidistant at △ z L/n, △ z is the equidistant distance between adjacent nodes, each node has one degree of translational freedom in the x-direction, and the lumped mass system shown in the model has one degree of translational freedom under dual-frequency excitationThe non-linear second order differential equation represented by a dot symbol is described as follows:
Figure FDA0002424596660000014
wherein M is a total mass matrix, D is a Rayleigh damping coefficient, A is a pneumatic damping matrix, K is a stiffness matrix, f1And f2Is the first and second node excitation force amplitude vector, omega1And omega2The angular eigenfrequencies of the first and second modes, x is the displacement vector, t is the time,
Figure FDA0002424596660000015
Figure FDA0002424596660000016
is the absolute value of the component velocity vector, wherein,
Figure FDA0002424596660000017
is the velocity vector of the first component,
Figure FDA0002424596660000018
is the absolute value velocity vector of the first component,
Figure FDA0002424596660000019
is the velocity vector of the nth component,
Figure FDA00024245966600000110
is the absolute value velocity vector of the nth component,
Figure FDA00024245966600000111
meaning a velocity vector of 2,3 … … components,
Figure FDA00024245966600000112
in the form of a velocity vector, the velocity vector,
Figure FDA00024245966600000113
is an acceleration vector;
step 2: solving the stiffness matrix by using the Euler-Bernoulli theorem, and performing semi-analytic processing on each element of an upper triangle of the symmetrical flexibility matrix by using a first-order Euler-Bernoulli beam theory according to an equation (2), wherein z is a wingspan coordinate from the base, and z is a wingspan coordinateiAnd z andjspanwise coordinates for the ith and j nodes, zi﹤zjUsing the principle of imaginary work, equation (2) represents the deformation of the node i caused by the unit load applied by the node j, equation (2) is obtained by numerical integration using the Simpson's rule, and the inverse matrix of the compliance matrix can give the stiffness matrix shown in equation ⑶,
Figure FDA0002424596660000021
K＝S-1(3)
in the formula, SijIs a compliance matrix element, S is a compliance matrix,
and step 3:
solving the total mass matrix, basic mass matrix elements, using a correlation function
Figure FDA00024245966600000211
Can be obtained by the following formula:
Figure FDA0002424596660000022
in the formula (I), the compound is shown in the specification,ijfor the Kronecker function, the lumped mass model formula requires that the tuning or excitation mass be distributed discretely over its nodes by distributing the kth tuning mass between its two neighboring nodes i and i +1
Figure FDA0002424596660000023
Equation (5) is used to calculate the corresponding distribution quality matrix
Figure FDA0002424596660000024
Figure FDA0002424596660000025
In the formula, zkFor the coordinates of the kth additional mass,
Figure FDA0002424596660000026
and
Figure FDA0002424596660000027
is an n-dimensional orthogonal basis vector,
Figure FDA0002424596660000028
and
Figure FDA0002424596660000029
is a transposed matrix of the i-dimensional orthogonal basis vectors,
Figure FDA00024245966600000210
for the kth spanwise coordinate, the total mass matrix M can thus be obtained:
Figure FDA0002424596660000031
in the formula (I), the compound is shown in the specification,
Figure FDA0002424596660000032
in order to be a basic quality matrix,
Figure FDA0002424596660000033
for the kth additional tuning or excitation quality matrix, ntIn order to tune the mass number,
and 4, step 4:
the influence of air damping, which is taken into account in the calculation, is expressed in its most common form as rayleigh damping, as shown in detail below,
D＝aM+bK (7)
in the above formula, a is 0 and b is 0.0156, a and b are rayleigh damping coefficients,
element A of the aerodynamic damping matrixijCan be obtained by the formula (8) wherein C is perpendicular to the flow of the flat platedAs a conservative approximation representing a complex aerodynamic situation:
Figure FDA0002424596660000034
where ρ is the air density at ambient room temperature, C (z) is the chord length, CdIs an aerodynamic drag coefficient of an angle of attack of 90 degrees,
Figure FDA0002424596660000035
Figure FDA0002424596660000036
by solving the generalized eigenvalue problem set forth in equation (9), the corresponding eigenvectors of the defined model shape are derived from equation (10), where φiN-dimensional eigenmode vectors for the ith mode:
and 5:
considering the influence of the static bending moment, ensuring higher reliability of the solving process, calculating additional parameters, calculating the static bending moment distribution caused by the self weight of the blade by a two-step procedure, and firstly calculating the shearing force Q (z) by using the equation (11) when the boundary condition shearing force Q (L) of the tip is 0:
Figure FDA0002424596660000041
where g is the gravitational acceleration, m(s) is the blade mass per unit length, then the moment distribution Ms (z) is obtained using equation (12), and the boundary condition Ms (L) is 0:
Figure FDA0002424596660000042
where Ms (z) is the bending moment of sway due to self weight, Ms (L) is the bending moment of sway due to self weight at length L, q(s) is the shear force at length s, the static bending moment distribution due to self weight of the external mass can be obtained by equation (13), and the resulting static bending moment can be obtained by equation (14) using the principle of superposition:
Figure FDA0002424596660000043
Figure FDA0002424596660000044
in the formula (I), the compound is shown in the specification,
Figure FDA0002424596660000045
for the k-th tuning mass,
Figure FDA0002424596660000046
spanwise coordinate of the kth additional mass, Mt(z) is the bending moment created by the tuned mass,
Figure FDA0002424596660000047
is static bending moment, niIs the maximum value of k, indicating the kth tuning quality
Figure FDA0002424596660000048
Has a minimum value of 1 and a maximum value of niThe node bending moment history M (z, t) used for damage calculation is represented by equation (15), where the node curvature is approximated by the central difference of the second derivative of the node displacement history x (z, t):
Figure FDA0002424596660000049
step 6:
solving the degree of mixing of the double excitation modes, and limiting the peak strain amplitude to experienceSensor type specific threshold value, root mean square strain threshold value given by equation (16)
Figure FDA0002424596660000051
Used as constraints in the optimization process discussed later:
Figure FDA0002424596660000052
wherein the time span T2-T1The strain history (z, t) of the end fiber corresponding to the steady-state portion of the time history and the flange, which yields bernoulli beams through the strain-moment relationship of first order Euler, as shown in equation (17):
Figure FDA0002424596660000053
in the formula, xc(z) is the distance between the terminal optical fiber and the bending axis;
in the case of dual-frequency excitation, the ratio between the two force amplitudes is measured by the modal mixing degree, defined here as
Figure FDA0002424596660000054
Where ψ is a degree of mixing of the two excitation modes, where ψ ═ 1 denotes F since there are two excitations1Far greater than F2Time-domain excitation, also known as pure mode-1 excitation; psi-0 denotes F2Excitation at 0, also called pure mode-2 excitation, F1And F2In order to obtain the amplitude of the exciting force,
and 7:
in the multi-frequency fatigue test method, a relational expression of design vector variables and related parameters is established, and the nonlinear constraint of final fatigue failure is established from the angle of energy and actual damage distribution;
in the dual-frequency method, the design variable vector q consists of two excitation forces: excitation position and scale factor, given by equation (19):
Figure FDA0002424596660000055
where w is a scale factor, the vector of the total node cycle number and the total test time and total dissipated energy are given by equations (20) to (24), and the objective function of the multi-frequency method conforms to equation (25), the node cycle number in the case of multi-frequency varies with the blade variation, and therefore, the non-linear constraint in equation (28) requires that the minimum node cycle number be equal to or greater than 3X106And (3) cyclic reference:
Figure FDA0002424596660000061
Ttot＝w(T2-T1) (21)
Figure FDA0002424596660000062
Figure FDA0002424596660000063
Figure FDA0002424596660000064
Figure FDA0002424596660000065
in the formula, NrsIs the number of cycles in unit rs, ntotAs vector of total node cycle number, TtotFor total test time, EtotFor total energy consumption, D is the actual injury index, EaerFor aerodynamic energy dissipation, EstrFor structural energy dissipation, DtFor the target damage index, f (q) is the algorithm optimization coefficient when the design variable vector is q, Dt(zi) Is a spanwise coordinate of ziTarget injury index, D (z)i) Is a spanwise coordinate of ziReal injury fingerThe number of the first and second groups is,
minimum f (q) obedience
3×106-ntot(q)＝0 (26)
Figure FDA0002424596660000066
Figure FDA0002424596660000067
In the formula (I), the compound is shown in the specification,
Figure FDA0002424596660000071
root mean square strain value, x, of time history100(q) is the 100 th displacement vector in the design of the variable vector q, x100The distance between the end fiber and the bending axis was 100 mm.
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